Vol. 19, No. 1, 1966

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An asymptotic property of the Euler function

Ronald Joseph Miech

Vol. 19 (1966), No. 1, 95–107

Let φ(n) denote the Euler function. The starting point of this paper is the simple observation that if p is a prime then p and φ(p) + 1 = p have a common divisor which is greater than 1; its conclusion is: if {mk} is the sequence of positive square free integers which have k prime factors, where k 2, then the number of integers mk not exceeding x such that mk and φ(mk) + 1 have a common divisor other than 1 is asymptotic to

λk-x--(log logx)k−2,
log x

where λk is a positive constant that depends on k.

The source of the problem under consideration was a question raised by Gordon in the course of his investigations of Hajós factorization of abelian groups. The question was: are there integers n, other than primes and their doubles, such that φ(n) + 1 divides n. This is still an open problem. However, if we relax our demands, as we have done above, it is possible to prove the asymptotic relation stated there.

Mathematical Subject Classification
Primary: 10.43
Received: 25 August 1965
Published: 1 October 1966
Ronald Joseph Miech